How to move one edge's vertice without interrupting other mesh
Connect Vertex Path &amp; Dissolve Edge
You don't have to move any vertices: they all are right in place. What you want is to re-arrange the topology, aka changing the connections between them. You'll need this two tools:
Use Connect Vertex Path to create an edge between two vertex

This tool connects vertices in the order they are selected, splitting
the faces between them. When there are only two vertices selected, a
cut will be made across unselected faces, a bit like the Knife tool;
but this is limited to straight cuts across connected faces.

https://docs.blender.org/manual/en/dev/modeling/meshes/editing/vertex/connect_vertex_path.html

and then Dissolve edges to remove an existing edge:

Removes edges sharing two faces (joining those faces).

https://docs.blender.org/manual/en/dev/modeling/meshes/editing/mesh/delete.html